You guys raised some good points on hunting and gun ownership traditions' effect on snipers.
Lets not forget that the fathers of the modern military sniper were Scottish hunters who employed their tradecraft on the battlefields of WWI.
While the military has spent a lot of time and resources developing the art in the decades since, a lot of the innovation and development comes from individual snipers themselves. Many of the best sniper rifle brands were created by keen shooters. These people have an advantage over scientists in that they have a lifetime of experience to draw on, whereas a scientist will be relying on graphs and computer models.
In many western countries, especially the US and Canada, gun laws are pretty lax and a lot of people literally grown up with guns. OTOH, most Chinese will only get their first look at a gun during a short period of military training at the start of high school.
Western Snipers will also get the chance to refine and practice their skills at home when off duty, whereas Chinese soldiers will get no such option since private gun ownership is banned in China.
Coming 3rd place is an amazing accomplishment when you take in the full picture. However, I do not think China will be able to design top class sniper rifles until they start involving snipers in the design process from the very start, much like what was done with fighter design.
